MinT - Best Known (85−52, 85, s)-Nets in Base 3

Best Known (85−52, 85, s)-Nets in Base 3

(85−52, 85, 38)-Net over F₃ — Constructive and digital

Digital (33, 85, 38)-net over F₃, using

t-expansion [i] based on digital (32, 85, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(85−52, 85, 46)-Net over F₃ — Digital

Digital (33, 85, 46)-net over F₃, using

net from sequence [i] based on digital (33, 45)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
  - a curve from the manYPoints database [i]

(85−52, 85, 138)-Net in Base 3 — Upper bound on s

There is no (33, 85, 139)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁸⁵, 139, S₃, 52), but
- the linear programming bound shows that MÂ â‰¥ 25187 471495 546450 938829 166628 945306 291038 411946 137194 049987 820699 630759 367931 165327 613809 362902 650627 817783 270689 253835 813240 163063 / 655379 887178 349656 963856 190956 437056 740438 060960 854359 735199 515832 284477 226278 150220 800000 > 3⁸⁵ [i]